Eye of the Tyger: early-time resonances and singularities in the inviscid Burgers equation

TL;DR

This paper uncovers early-time complex singularities in the inviscid Burgers equation, revealing their physical relevance and proposing methods to mitigate associated numerical artifacts called tygers.

Contribution

It identifies and analyzes early-time resonances and singularities in the inviscid Burgers equation, introduces a novel singularity theory, and proposes techniques to reduce tygers in simulations.

Findings

Complex singularities form an eye shape in the time domain.

Loss of convergence occurs at about 2/3 of the pre-shock time.

Methods like tyger purging and UV completion mitigate early-time tygers.

Abstract

We chart a singular landscape in the temporal domain of the inviscid Burgers equation in one space dimension for sine-wave initial conditions. These so far undetected complex singularities are arranged in an eye shape centered around the origin in time. Interestingly, since the eye is squashed along the imaginary time axis, complex-time singularities can become physically relevant at times well before the first real singularity -- the pre-shock. Indeed, employing a time-Taylor representation for the velocity around $t=0$, loss of convergence occurs roughly at 2/3 of the pre-shock time for the considered single- and multi-mode models. Furthermore, the loss of convergence is accompanied by the appearance of initially localized resonant behaviour which, as we claim, is a temporal manifestation of the so-called tyger phenomenon, reported in Galerkin-truncated implementations of inviscid fluids [Ray et al., Phys. Rev. E 84, 016301 (2011)]. We support our findings of early-time tygers with two complementary and independent means, namely by an asymptotic analysis of the time-Taylor series for the velocity, as well as by a novel singularity theory that employs Lagrangian coordinates. Finally, we apply two methods that reduce the amplitude of early-time tygers, one is tyger purging which removes large Fourier modes from the velocity, and is a variant of a procedure known in the literature. The other method realizes an iterative UV completion, which, most interestingly, iteratively restores the conservation of energy once the Taylor series for the velocity diverges. Our techniques are straightforwardly adapted to higher dimensions and/or applied to other equations of hydrodynamics.